The Study Of Existence Of Positive Solutions For Second-Order Impulsive Differential Equations

In this paper,we mainly study the existence of positive solutions for three classes of second-order impulsive functional differential equations.Firstly,by using Leggett-Williams’ fixed point theorem,the existence of three positive solutions for the second-order impulsive differential equations with integral boundary conditions and its applications is considered.The approximate range of three solutions is given.And we give an example to verify our results.Secondly,Applying the eigenvalue theory and the theory of α-concave operator,we establish some new sufficient conditions to guarantee the existence and continuity of positive solutions on a parameter for a second-order impulsive differential equation,two nonexistence results of positive solutions are also given.In particular,we prove that the unique solution of the problem is strongly increasing and depends continuously on the parameter.And we give two examples to verify the reasonability of our results.Finally,by using a transformation technique to deal with the impulse term of a second-order singular impulsive differential equation,the fixed point theorem in a cone in Banach space and H(?)lder’s inequality,this paper establishes sharp conditions to guarantee the existence of the first order derivative positive solutions.The results significantly extend and improve many known results for both the continuous case and more general nonsingular case.